Question: Is ${131904}$ divisible by $9$ ?
Answer: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {131904}= &&{1}\cdot100000+ \\&&{3}\cdot10000+ \\&&{1}\cdot1000+ \\&&{9}\cdot100+ \\&&{0}\cdot10+ \\&&{4}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {131904}= &&{1}(99999+1)+ \\&&{3}(9999+1)+ \\&&{1}(999+1)+ \\&&{9}(99+1)+ \\&&{0}(9+1)+ \\&&{4} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {131904}= &&\gray{1\cdot99999}+ \\&&\gray{3\cdot9999}+ \\&&\gray{1\cdot999}+ \\&&\gray{9\cdot99}+ \\&&\gray{0\cdot9}+ \\&& {1}+{3}+{1}+{9}+{0}+{4} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${131904}$ is divisible by $9$ if ${ 1}+{3}+{1}+{9}+{0}+{4}$ is divisible by $9$ Add the digits of ${131904}$ $ {1}+{3}+{1}+{9}+{0}+{4} = {18} $ If ${18}$ is divisible by $9$ , then ${131904}$ must also be divisible by $9$ ${18}$ is divisible by $9$, therefore ${131904}$ must also be divisible by $9$.